astronomy

James Ax

James Ax is a astronomy topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand James Ax rather than just read about it. In short: James Burton Ax (10 January 1937 – 11 June 2006) was an American mathematician who made contributions in algebra and number theory using model theory. He shared, with Simon B.

Key takeaways

James Ax belongs to astronomy; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect James Ax to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of James Ax from memory before moving on to harder problems.

Reference excerpt

James Burton Ax (10 January 1937 – 11 June 2006) was an American mathematician who made contributions in algebra and number theory using model theory. He shared, with Simon B. Kochen, the seventh Frank Nelson Cole Prize in Number Theory, which was awarded for a series of three joint papers on Diophantine problems.

Education and career Ax was born in New York City and graduated from Stuyvesant High School in 1954. He then joined the Brooklyn Polytechnic University. He earned his Ph.D. from the University of California, Berkeley in 1961 under the direction of Gerhard Hochschild, with a dissertation on The Intersection of Norm Groups. After a year at Stanford University, he joined the mathematics faculty at Cornell University. He spent the academic year 1965–1966 at Harvard University on a Guggenheim Fellowship. In 1969, he was recruited by his Berkeley classmate Jim Simons to move from Cornell to the mathematics department at Stony Brook University. In 1970 he was an Invited Speaker at the ICM in Nice with the talk Transcendence and differential algebraic geometry. In the 1970s, he worked on the fundamentals of physics, including an axiomatization of space-time and the group theoretical properties of the axioms of quantum mechanics. In 1977 he retired from his academic career and joined a hedge fund run by Jim Simons. In the 1980s, he and Simons founded the quantitative finance firm Axcom Trading Advisors, which was later acquired by Renaissance Technologies and renamed the Medallion Fund. The latter fund was named after the Cole Prize won by James Ax and the Veblen Prize won by Jim Simons. In the early 1990s, Ax retired from his financial career and went to San Diego, California, where he studied further on the foundations of quantum mechanics and also attended, at the University of California, San Diego, courses on playwriting and screenwriting. (In 2005 he completed a thriller screenplay entitled Bots.) The Ax Library in the Department of Mathematics at the University of California, San Diego houses his mathematical books.

Personal Ax is the father of American cosmologist Brian Keating and Kevin B. Keating (b. 1967), who is the president of the Kevin and Masha Keating Family Foundation. After Ax and his first wife divorced, she remarried a man named Keating, and young Brian and his older brother Kevin took the stepfather's name. Brian Keating explained (in 2020) that he and his father were not close during his childhood; his father often joked that, "I don't really care about kids until they learn algebra."

Selected publications Ax, James (1962). "The intersection of norm groups". Transactions of the American Mathematical Society. 105 (3): 462–474. doi:10.1090/s0002-9947-1962-0141651-2. MR 0141651. Ax, James (1965). "A field of cohomological dimension 1 which is not C1". Bulletin of the American Mathematical Society. 71 (5): 717–718. doi:10.1090/s0002-9904-1965-11354-4. MR 0199177. Ax, James (1965). "On the undecidability of power series fields". Proceedings of the American Mathematical Society. 16 (4): 846. doi:10.1090/s0002-9939-1965-0177890-2. MR 0177890. Ax, James (1965). "Proof of some conjectures on cohomological dimension". Proceedings of the American Mathematical Society. 16 (6): 1214–1221. doi:10.1090/s0002-9939-1965-0188263-0. MR 0188263. Ax, James (1965). "On the units of an algebraic number field". Illinois Journal of Mathematics. 9 (4): 584–589. doi:10.1215/ijm/1256059299. MR 0181630. Ax, James (1967). "Solving diophantine problems modulo every prime". Annals of Mathematics. Series 2. 85 (2): 161–183. doi:10.2307/1970438. JSTOR 1970438. Ax, James (1968). "The elementary theory of finite fields". Annals of Mathematics. Series 2. 88 (2): 239–271. doi:10.2307/1970573. JSTOR 1970573. Ax, James (1970). "Zeros of polynomials over local fields—The Galois action". Journal of Algebra. 15 (3): 417–428. doi:10.1016/0021-8693(70)90069-4. Ax, James (1971). "On Schanuel's conjectures". Annals of Mathematics. Series 2. 93 (2): 252–268. doi:10.2307/1970774. JSTOR 1970774.

See also Leopoldt's conjecture Schanuel's conjecture

References

External links James Ax at the Mathematics Genealogy Project James B. Ax Library - at UCSD.

Worked examples

Example 1 — a first encounter with James Ax

Start with the simplest possible case. Write down what James Ax claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In astronomy, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to James Ax before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about James Ax ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of James Ax

In research: James Ax appears in astronomy research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses James Ax in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: James Ax is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1937 births, 2006 deaths, American number theorists, so understanding it makes those chapters shorter.
In everyday life: Look for James Ax outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study James Ax in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what James Ax means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain James Ax out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is James Ax in simple terms?

James Burton Ax (10 January 1937 – 11 June 2006) was an American mathematician who made contributions in algebra and number theory using model theory. He shared, with Simon B.

Why does James Ax matter?

Because it connects several astronomy ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study James Ax?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on James Ax.